3.7.23 \(\int \frac {1}{(d+e x)^{5/2} (a+c x^2)} \, dx\) [623]
Optimal. Leaf size=736 \[ -\frac {2 e}{3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}-\frac {4 c d e}{\left (c d^2+a e^2\right )^2 \sqrt {d+e x}}+\frac {c^{3/4} e \left (3 c d^2-a e^2-2 \sqrt {c} d \sqrt {c d^2+a e^2}\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {2} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {c^{3/4} e \left (3 c d^2-a e^2-2 \sqrt {c} d \sqrt {c d^2+a e^2}\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {2} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {c^{3/4} e \left (3 c d^2-a e^2+2 \sqrt {c} d \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {c^{3/4} e \left (3 c d^2-a e^2+2 \sqrt {c} d \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}} \]
[Out]
-2/3*e/(a*e^2+c*d^2)/(e*x+d)^(3/2)-4*c*d*e/(a*e^2+c*d^2)^2/(e*x+d)^(1/2)+1/2*c^(3/4)*e*arctanh((-c^(1/4)*2^(1/
2)*(e*x+d)^(1/2)+(d*c^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2))/(d*c^(1/2)-(a*e^2+c*d^2)^(1/2))^(1/2))*(3*c*d^2-a*e^2-
2*d*c^(1/2)*(a*e^2+c*d^2)^(1/2))/(a*e^2+c*d^2)^(5/2)*2^(1/2)/(d*c^(1/2)-(a*e^2+c*d^2)^(1/2))^(1/2)-1/2*c^(3/4)
*e*arctanh((c^(1/4)*2^(1/2)*(e*x+d)^(1/2)+(d*c^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2))/(d*c^(1/2)-(a*e^2+c*d^2)^(1/2
))^(1/2))*(3*c*d^2-a*e^2-2*d*c^(1/2)*(a*e^2+c*d^2)^(1/2))/(a*e^2+c*d^2)^(5/2)*2^(1/2)/(d*c^(1/2)-(a*e^2+c*d^2)
^(1/2))^(1/2)-1/4*c^(3/4)*e*ln((e*x+d)*c^(1/2)+(a*e^2+c*d^2)^(1/2)-c^(1/4)*2^(1/2)*(e*x+d)^(1/2)*(d*c^(1/2)+(a
*e^2+c*d^2)^(1/2))^(1/2))*(3*c*d^2-a*e^2+2*d*c^(1/2)*(a*e^2+c*d^2)^(1/2))/(a*e^2+c*d^2)^(5/2)*2^(1/2)/(d*c^(1/
2)+(a*e^2+c*d^2)^(1/2))^(1/2)+1/4*c^(3/4)*e*ln((e*x+d)*c^(1/2)+(a*e^2+c*d^2)^(1/2)+c^(1/4)*2^(1/2)*(e*x+d)^(1/
2)*(d*c^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2))*(3*c*d^2-a*e^2+2*d*c^(1/2)*(a*e^2+c*d^2)^(1/2))/(a*e^2+c*d^2)^(5/2)*
2^(1/2)/(d*c^(1/2)+(a*e^2+c*d^2)^(1/2))^(1/2)
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Rubi [A]
time = 1.13, antiderivative size = 736, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 8, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {724, 843, 841,
1183, 648, 632, 212, 642} \begin {gather*} -\frac {c^{3/4} e \left (2 \sqrt {c} d \sqrt {a e^2+c d^2}-a e^2+3 c d^2\right ) \log \left (-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {a e^2+c d^2}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} \left (a e^2+c d^2\right )^{5/2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {c^{3/4} e \left (2 \sqrt {c} d \sqrt {a e^2+c d^2}-a e^2+3 c d^2\right ) \log \left (\sqrt {2} \sqrt [4]{c} \sqrt {d+e x} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {a e^2+c d^2}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} \left (a e^2+c d^2\right )^{5/2} \sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}}+\frac {c^{3/4} e \left (-2 \sqrt {c} d \sqrt {a e^2+c d^2}-a e^2+3 c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}-\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}\right )}{\sqrt {2} \left (a e^2+c d^2\right )^{5/2} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}-\frac {c^{3/4} e \left (-2 \sqrt {c} d \sqrt {a e^2+c d^2}-a e^2+3 c d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a e^2+c d^2}+\sqrt {c} d}+\sqrt {2} \sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}\right )}{\sqrt {2} \left (a e^2+c d^2\right )^{5/2} \sqrt {\sqrt {c} d-\sqrt {a e^2+c d^2}}}-\frac {4 c d e}{\sqrt {d+e x} \left (a e^2+c d^2\right )^2}-\frac {2 e}{3 (d+e x)^{3/2} \left (a e^2+c d^2\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[1/((d + e*x)^(5/2)*(a + c*x^2)),x]
[Out]
(-2*e)/(3*(c*d^2 + a*e^2)*(d + e*x)^(3/2)) - (4*c*d*e)/((c*d^2 + a*e^2)^2*Sqrt[d + e*x]) + (c^(3/4)*e*(3*c*d^2
- a*e^2 - 2*Sqrt[c]*d*Sqrt[c*d^2 + a*e^2])*ArcTanh[(Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]] - Sqrt[2]*c^(1/4)*S
qrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(Sqrt[2]*(c*d^2 + a*e^2)^(5/2)*Sqrt[Sqrt[c]*d - Sqrt[c*d
^2 + a*e^2]]) - (c^(3/4)*e*(3*c*d^2 - a*e^2 - 2*Sqrt[c]*d*Sqrt[c*d^2 + a*e^2])*ArcTanh[(Sqrt[Sqrt[c]*d + Sqrt[
c*d^2 + a*e^2]] + Sqrt[2]*c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]])/(Sqrt[2]*(c*d^2 + a*e
^2)^(5/2)*Sqrt[Sqrt[c]*d - Sqrt[c*d^2 + a*e^2]]) - (c^(3/4)*e*(3*c*d^2 - a*e^2 + 2*Sqrt[c]*d*Sqrt[c*d^2 + a*e^
2])*Log[Sqrt[c*d^2 + a*e^2] - Sqrt[2]*c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]*Sqrt[d + e*x] + Sqrt[c]*(d
+ e*x)])/(2*Sqrt[2]*(c*d^2 + a*e^2)^(5/2)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^2]]) + (c^(3/4)*e*(3*c*d^2 - a*e^
2 + 2*Sqrt[c]*d*Sqrt[c*d^2 + a*e^2])*Log[Sqrt[c*d^2 + a*e^2] + Sqrt[2]*c^(1/4)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a
*e^2]]*Sqrt[d + e*x] + Sqrt[c]*(d + e*x)])/(2*Sqrt[2]*(c*d^2 + a*e^2)^(5/2)*Sqrt[Sqrt[c]*d + Sqrt[c*d^2 + a*e^
2]])
Rule 212
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 632
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 642
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Rule 648
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && !NiceSqrtQ[b^2 - 4*a*c]
Rule 724
Int[((d_) + (e_.)*(x_))^(m_)/((a_) + (c_.)*(x_)^2), x_Symbol] :> Simp[e*((d + e*x)^(m + 1)/((m + 1)*(c*d^2 + a
*e^2))), x] + Dist[c/(c*d^2 + a*e^2), Int[(d + e*x)^(m + 1)*((d - e*x)/(a + c*x^2)), x], x] /; FreeQ[{a, c, d,
e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1]
Rule 841
Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2, Subst[Int[(e*f
- d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0]
Rule 843
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*f - d*g)*((d
+ e*x)^(m + 1)/((m + 1)*(c*d^2 + a*e^2))), x] + Dist[1/(c*d^2 + a*e^2), Int[(d + e*x)^(m + 1)*(Simp[c*d*f + a*
e*g - c*(e*f - d*g)*x, x]/(a + c*x^2)), x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && NeQ[c*d^2 + a*e^2, 0] &&
FractionQ[m] && LtQ[m, -1]
Rule 1183
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r =
Rt[2*q - b/c, 2]}, Dist[1/(2*c*q*r), Int[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(
d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^{5/2} \left (a+c x^2\right )} \, dx &=-\frac {2 e}{3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}+\frac {c \int \frac {d-e x}{(d+e x)^{3/2} \left (a+c x^2\right )} \, dx}{c d^2+a e^2}\\ &=-\frac {2 e}{3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}-\frac {4 c d e}{\left (c d^2+a e^2\right )^2 \sqrt {d+e x}}+\frac {c \int \frac {c d^2-a e^2-2 c d e x}{\sqrt {d+e x} \left (a+c x^2\right )} \, dx}{\left (c d^2+a e^2\right )^2}\\ &=-\frac {2 e}{3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}-\frac {4 c d e}{\left (c d^2+a e^2\right )^2 \sqrt {d+e x}}+\frac {(2 c) \text {Subst}\left (\int \frac {2 c d^2 e+e \left (c d^2-a e^2\right )-2 c d e x^2}{c d^2+a e^2-2 c d x^2+c x^4} \, dx,x,\sqrt {d+e x}\right )}{\left (c d^2+a e^2\right )^2}\\ &=-\frac {2 e}{3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}-\frac {4 c d e}{\left (c d^2+a e^2\right )^2 \sqrt {d+e x}}+\frac {c^{3/4} \text {Subst}\left (\int \frac {\frac {\sqrt {2} \left (2 c d^2 e+e \left (c d^2-a e^2\right )\right ) \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}-\left (2 c d^2 e+e \left (c d^2-a e^2\right )+2 \sqrt {c} d e \sqrt {c d^2+a e^2}\right ) x}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{\sqrt {2} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {c^{3/4} \text {Subst}\left (\int \frac {\frac {\sqrt {2} \left (2 c d^2 e+e \left (c d^2-a e^2\right )\right ) \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+\left (2 c d^2 e+e \left (c d^2-a e^2\right )+2 \sqrt {c} d e \sqrt {c d^2+a e^2}\right ) x}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{\sqrt {2} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\\ &=-\frac {2 e}{3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}-\frac {4 c d e}{\left (c d^2+a e^2\right )^2 \sqrt {d+e x}}+\frac {\left (\sqrt {c} e \left (3 c d^2-a e^2-2 \sqrt {c} d \sqrt {c d^2+a e^2}\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{2 \left (c d^2+a e^2\right )^{5/2}}+\frac {\left (\sqrt {c} e \left (3 c d^2-a e^2-2 \sqrt {c} d \sqrt {c d^2+a e^2}\right )\right ) \text {Subst}\left (\int \frac {1}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{2 \left (c d^2+a e^2\right )^{5/2}}-\frac {\left (c^{3/4} e \left (3 c d^2-a e^2+2 \sqrt {c} d \sqrt {c d^2+a e^2}\right )\right ) \text {Subst}\left (\int \frac {-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 x}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{2 \sqrt {2} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {\left (c^{3/4} e \left (3 c d^2-a e^2+2 \sqrt {c} d \sqrt {c d^2+a e^2}\right )\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 x}{\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} x}{\sqrt [4]{c}}+x^2} \, dx,x,\sqrt {d+e x}\right )}{2 \sqrt {2} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\\ &=-\frac {2 e}{3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}-\frac {4 c d e}{\left (c d^2+a e^2\right )^2 \sqrt {d+e x}}-\frac {c^{3/4} e \left (3 c d^2-a e^2+2 \sqrt {c} d \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {c^{3/4} e \left (3 c d^2-a e^2+2 \sqrt {c} d \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}-\frac {\left (\sqrt {c} e \left (3 c d^2-a e^2-2 \sqrt {c} d \sqrt {c d^2+a e^2}\right )\right ) \text {Subst}\left (\int \frac {1}{2 \left (d-\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right )-x^2} \, dx,x,-\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{\left (c d^2+a e^2\right )^{5/2}}-\frac {\left (\sqrt {c} e \left (3 c d^2-a e^2-2 \sqrt {c} d \sqrt {c d^2+a e^2}\right )\right ) \text {Subst}\left (\int \frac {1}{2 \left (d-\frac {\sqrt {c d^2+a e^2}}{\sqrt {c}}\right )-x^2} \, dx,x,\frac {\sqrt {2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+2 \sqrt {d+e x}\right )}{\left (c d^2+a e^2\right )^{5/2}}\\ &=-\frac {2 e}{3 \left (c d^2+a e^2\right ) (d+e x)^{3/2}}-\frac {4 c d e}{\left (c d^2+a e^2\right )^2 \sqrt {d+e x}}+\frac {c^{3/4} e \left (3 c d^2-a e^2-2 \sqrt {c} d \sqrt {c d^2+a e^2}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}-\sqrt {2} \sqrt {d+e x}\right )}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {2} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {c^{3/4} e \left (3 c d^2-a e^2-2 \sqrt {c} d \sqrt {c d^2+a e^2}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{c} \left (\frac {\sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}{\sqrt [4]{c}}+\sqrt {2} \sqrt {d+e x}\right )}{\sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}\right )}{\sqrt {2} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d-\sqrt {c d^2+a e^2}}}-\frac {c^{3/4} e \left (3 c d^2-a e^2+2 \sqrt {c} d \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}-\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}+\frac {c^{3/4} e \left (3 c d^2-a e^2+2 \sqrt {c} d \sqrt {c d^2+a e^2}\right ) \log \left (\sqrt {c d^2+a e^2}+\sqrt {2} \sqrt [4]{c} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}} \sqrt {d+e x}+\sqrt {c} (d+e x)\right )}{2 \sqrt {2} \left (c d^2+a e^2\right )^{5/2} \sqrt {\sqrt {c} d+\sqrt {c d^2+a e^2}}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.82, size = 265, normalized size = 0.36 \begin {gather*} -\frac {2 e \left (a e^2+c d (7 d+6 e x)\right )}{3 \left (c d^2+a e^2\right )^2 (d+e x)^{3/2}}+\frac {\sqrt {c} \sqrt {-c d-i \sqrt {a} \sqrt {c} e} \tan ^{-1}\left (\frac {\sqrt {-c d-i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+i \sqrt {a} e}\right )}{\sqrt {a} \left (-i \sqrt {c} d+\sqrt {a} e\right )^3}+\frac {\sqrt {c} \sqrt {-c d+i \sqrt {a} \sqrt {c} e} \tan ^{-1}\left (\frac {\sqrt {-c d+i \sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-i \sqrt {a} e}\right )}{\sqrt {a} \left (i \sqrt {c} d+\sqrt {a} e\right )^3} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[1/((d + e*x)^(5/2)*(a + c*x^2)),x]
[Out]
(-2*e*(a*e^2 + c*d*(7*d + 6*e*x)))/(3*(c*d^2 + a*e^2)^2*(d + e*x)^(3/2)) + (Sqrt[c]*Sqrt[-(c*d) - I*Sqrt[a]*Sq
rt[c]*e]*ArcTan[(Sqrt[-(c*d) - I*Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d + I*Sqrt[a]*e)])/(Sqrt[a]*((-I)*
Sqrt[c]*d + Sqrt[a]*e)^3) + (Sqrt[c]*Sqrt[-(c*d) + I*Sqrt[a]*Sqrt[c]*e]*ArcTan[(Sqrt[-(c*d) + I*Sqrt[a]*Sqrt[c
]*e]*Sqrt[d + e*x])/(Sqrt[c]*d - I*Sqrt[a]*e)])/(Sqrt[a]*(I*Sqrt[c]*d + Sqrt[a]*e)^3)
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2546\) vs.
\(2(595)=1190\).
time = 0.62, size = 2547, normalized size = 3.46
| | |
| method |
result |
size |
| | |
| derivativedivides |
\(\text {Expression too large to display}\) |
\(2547\) |
| default |
\(\text {Expression too large to display}\) |
\(2547\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(e*x+d)^(5/2)/(c*x^2+a),x,method=_RETURNVERBOSE)
[Out]
2*e*(-1/(a*e^2+c*d^2)^2*c*(1/4/a/c/e^2/(a*e^2+c*d^2)^(3/2)*(1/2*(3*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c^(
7/2)*d^5+2*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c^(5/2)*a*d^3*e^2-3*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)
*c^(5/2)*(a*c*e^2+c^2*d^2)^(1/2)*d^4-(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c^(3/2)*a^2*d*e^4-2*(2*(a*c*e^2+c
^2*d^2)^(1/2)+2*c*d)^(1/2)*c^(3/2)*(a*c*e^2+c^2*d^2)^(1/2)*a*d^2*e^2+(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c
^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*a^2*e^4+2*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*e^2+c*d^2)^(1/2)*a*c^2*d^2
*e^2+2*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*e^2+c*d^2)^(1/2)*c^3*d^4-2*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)
^(1/2)*(a*e^2+c*d^2)^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*a*c*d*e^2-2*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*e^2+
c*d^2)^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*c^2*d^3)/c^(1/2)*ln((e*x+d)*c^(1/2)+(e*x+d)^(1/2)*(2*((a*e^2+c*d^2)*c)^(1
/2)+2*c*d)^(1/2)+(a*e^2+c*d^2)^(1/2))+2*(2*e^6*c*a^3-4*d^2*e^4*a^2*c^2-6*d^4*e^2*c^3*a-1/2*(3*(2*(a*c*e^2+c^2*
d^2)^(1/2)+2*c*d)^(1/2)*c^(7/2)*d^5+2*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c^(5/2)*a*d^3*e^2-3*(2*(a*c*e^2+
c^2*d^2)^(1/2)+2*c*d)^(1/2)*c^(5/2)*(a*c*e^2+c^2*d^2)^(1/2)*d^4-(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c^(3/2
)*a^2*d*e^4-2*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c^(3/2)*(a*c*e^2+c^2*d^2)^(1/2)*a*d^2*e^2+(2*(a*c*e^2+c^
2*d^2)^(1/2)+2*c*d)^(1/2)*c^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*a^2*e^4+2*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a
*e^2+c*d^2)^(1/2)*a*c^2*d^2*e^2+2*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*e^2+c*d^2)^(1/2)*c^3*d^4-2*(2*(a*
c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*e^2+c*d^2)^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*a*c*d*e^2-2*(2*(a*c*e^2+c^2*d^2)
^(1/2)+2*c*d)^(1/2)*(a*e^2+c*d^2)^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*c^2*d^3)*(2*((a*e^2+c*d^2)*c)^(1/2)+2*c*d)^(1/
2)/c^(1/2))/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^(1/2)-2*c*d)^(1/2)*arctan((2*c^(1/2)*(e*x+d)^(1
/2)+(2*((a*e^2+c*d^2)*c)^(1/2)+2*c*d)^(1/2))/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^(1/2)-2*c*d)^(
1/2)))+1/4/a/c/e^2/(a*e^2+c*d^2)^(3/2)*(1/2*(-3*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c^(7/2)*d^5-2*(2*(a*c*
e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c^(5/2)*a*d^3*e^2+3*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c^(5/2)*(a*c*e^2+c
^2*d^2)^(1/2)*d^4+(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c^(3/2)*a^2*d*e^4+2*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d
)^(1/2)*c^(3/2)*(a*c*e^2+c^2*d^2)^(1/2)*a*d^2*e^2-(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c^(1/2)*(a*c*e^2+c^2
*d^2)^(1/2)*a^2*e^4-2*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*e^2+c*d^2)^(1/2)*a*c^2*d^2*e^2-2*(2*(a*c*e^2+
c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*e^2+c*d^2)^(1/2)*c^3*d^4+2*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*e^2+c*d^2
)^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*a*c*d*e^2+2*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*e^2+c*d^2)^(1/2)*(a*c*e
^2+c^2*d^2)^(1/2)*c^2*d^3)/c^(1/2)*ln((e*x+d)*c^(1/2)-(e*x+d)^(1/2)*(2*((a*e^2+c*d^2)*c)^(1/2)+2*c*d)^(1/2)+(a
*e^2+c*d^2)^(1/2))+2*(2*e^6*c*a^3-4*d^2*e^4*a^2*c^2-6*d^4*e^2*c^3*a+1/2*(-3*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^
(1/2)*c^(7/2)*d^5-2*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c^(5/2)*a*d^3*e^2+3*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c
*d)^(1/2)*c^(5/2)*(a*c*e^2+c^2*d^2)^(1/2)*d^4+(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c^(3/2)*a^2*d*e^4+2*(2*(
a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*c^(3/2)*(a*c*e^2+c^2*d^2)^(1/2)*a*d^2*e^2-(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d
)^(1/2)*c^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*a^2*e^4-2*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*e^2+c*d^2)^(1/2)*
a*c^2*d^2*e^2-2*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2)*(a*e^2+c*d^2)^(1/2)*c^3*d^4+2*(2*(a*c*e^2+c^2*d^2)^(1/
2)+2*c*d)^(1/2)*(a*e^2+c*d^2)^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*a*c*d*e^2+2*(2*(a*c*e^2+c^2*d^2)^(1/2)+2*c*d)^(1/2
)*(a*e^2+c*d^2)^(1/2)*(a*c*e^2+c^2*d^2)^(1/2)*c^2*d^3)*(2*((a*e^2+c*d^2)*c)^(1/2)+2*c*d)^(1/2)/c^(1/2))/(4*(a*
e^2+c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^(1/2)-2*c*d)^(1/2)*arctan((2*c^(1/2)*(e*x+d)^(1/2)-(2*((a*e^2+c*d
^2)*c)^(1/2)+2*c*d)^(1/2))/(4*(a*e^2+c*d^2)^(1/2)*c^(1/2)-2*((a*e^2+c*d^2)*c)^(1/2)-2*c*d)^(1/2))))-1/3/(a*e^2
+c*d^2)/(e*x+d)^(3/2)-2/(a*e^2+c*d^2)^2*c*d/(e*x+d)^(1/2))
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x+d)^(5/2)/(c*x^2+a),x, algorithm="maxima")
[Out]
integrate(1/((c*x^2 + a)*(x*e + d)^(5/2)), x)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 4828 vs.
\(2 (585) = 1170\).
time = 1.37, size = 4828, normalized size = 6.56 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x+d)^(5/2)/(c*x^2+a),x, algorithm="fricas")
[Out]
1/6*(3*(2*c^2*d^5*x*e + c^2*d^6 + 4*a*c*d^3*x*e^3 + a^2*x^2*e^6 + 2*a^2*d*x*e^5 + (2*a*c*d^2*x^2 + a^2*d^2)*e^
4 + (c^2*d^4*x^2 + 2*a*c*d^4)*e^2)*sqrt(-(c^4*d^5 - 10*a*c^3*d^3*e^2 + 5*a^2*c^2*d*e^4 + (a*c^5*d^10 + 5*a^2*c
^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 + 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 + a^6*e^10)*sqrt(-(25*c^7*d^8*e^2 - 100
*a*c^6*d^6*e^4 + 110*a^2*c^5*d^4*e^6 - 20*a^3*c^4*d^2*e^8 + a^4*c^3*e^10)/(a*c^10*d^20 + 10*a^2*c^9*d^18*e^2 +
45*a^3*c^8*d^16*e^4 + 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 + 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e
^12 + 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 + 10*a^10*c*d^2*e^18 + a^11*e^20)))/(a*c^5*d^10 + 5*a^2*c^4*d
^8*e^2 + 10*a^3*c^3*d^6*e^4 + 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 + a^6*e^10))*log((5*c^4*d^4*e - 10*a*c^3*d^
2*e^3 + a^2*c^2*e^5)*sqrt(x*e + d) + (15*a*c^4*d^6*e^2 - 35*a^2*c^3*d^4*e^4 + 13*a^3*c^2*d^2*e^6 - a^4*c*e^8 +
(a*c^6*d^13 + 2*a^2*c^5*d^11*e^2 - 5*a^3*c^4*d^9*e^4 - 20*a^4*c^3*d^7*e^6 - 25*a^5*c^2*d^5*e^8 - 14*a^6*c*d^3
*e^10 - 3*a^7*d*e^12)*sqrt(-(25*c^7*d^8*e^2 - 100*a*c^6*d^6*e^4 + 110*a^2*c^5*d^4*e^6 - 20*a^3*c^4*d^2*e^8 + a
^4*c^3*e^10)/(a*c^10*d^20 + 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 + 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^1
2*e^8 + 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 + 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 + 10*a^10*c*
d^2*e^18 + a^11*e^20)))*sqrt(-(c^4*d^5 - 10*a*c^3*d^3*e^2 + 5*a^2*c^2*d*e^4 + (a*c^5*d^10 + 5*a^2*c^4*d^8*e^2
+ 10*a^3*c^3*d^6*e^4 + 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 + a^6*e^10)*sqrt(-(25*c^7*d^8*e^2 - 100*a*c^6*d^6*
e^4 + 110*a^2*c^5*d^4*e^6 - 20*a^3*c^4*d^2*e^8 + a^4*c^3*e^10)/(a*c^10*d^20 + 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8
*d^16*e^4 + 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 + 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 + 120*a
^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 + 10*a^10*c*d^2*e^18 + a^11*e^20)))/(a*c^5*d^10 + 5*a^2*c^4*d^8*e^2 + 10
*a^3*c^3*d^6*e^4 + 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 + a^6*e^10))) - 3*(2*c^2*d^5*x*e + c^2*d^6 + 4*a*c*d^3
*x*e^3 + a^2*x^2*e^6 + 2*a^2*d*x*e^5 + (2*a*c*d^2*x^2 + a^2*d^2)*e^4 + (c^2*d^4*x^2 + 2*a*c*d^4)*e^2)*sqrt(-(c
^4*d^5 - 10*a*c^3*d^3*e^2 + 5*a^2*c^2*d*e^4 + (a*c^5*d^10 + 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 + 10*a^4*c^
2*d^4*e^6 + 5*a^5*c*d^2*e^8 + a^6*e^10)*sqrt(-(25*c^7*d^8*e^2 - 100*a*c^6*d^6*e^4 + 110*a^2*c^5*d^4*e^6 - 20*a
^3*c^4*d^2*e^8 + a^4*c^3*e^10)/(a*c^10*d^20 + 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 + 120*a^4*c^7*d^14*e^6
+ 210*a^5*c^6*d^12*e^8 + 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 + 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4
*e^16 + 10*a^10*c*d^2*e^18 + a^11*e^20)))/(a*c^5*d^10 + 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 + 10*a^4*c^2*d^
4*e^6 + 5*a^5*c*d^2*e^8 + a^6*e^10))*log((5*c^4*d^4*e - 10*a*c^3*d^2*e^3 + a^2*c^2*e^5)*sqrt(x*e + d) - (15*a*
c^4*d^6*e^2 - 35*a^2*c^3*d^4*e^4 + 13*a^3*c^2*d^2*e^6 - a^4*c*e^8 + (a*c^6*d^13 + 2*a^2*c^5*d^11*e^2 - 5*a^3*c
^4*d^9*e^4 - 20*a^4*c^3*d^7*e^6 - 25*a^5*c^2*d^5*e^8 - 14*a^6*c*d^3*e^10 - 3*a^7*d*e^12)*sqrt(-(25*c^7*d^8*e^2
- 100*a*c^6*d^6*e^4 + 110*a^2*c^5*d^4*e^6 - 20*a^3*c^4*d^2*e^8 + a^4*c^3*e^10)/(a*c^10*d^20 + 10*a^2*c^9*d^18
*e^2 + 45*a^3*c^8*d^16*e^4 + 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 + 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4
*d^8*e^12 + 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 + 10*a^10*c*d^2*e^18 + a^11*e^20)))*sqrt(-(c^4*d^5 - 10
*a*c^3*d^3*e^2 + 5*a^2*c^2*d*e^4 + (a*c^5*d^10 + 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 + 10*a^4*c^2*d^4*e^6 +
5*a^5*c*d^2*e^8 + a^6*e^10)*sqrt(-(25*c^7*d^8*e^2 - 100*a*c^6*d^6*e^4 + 110*a^2*c^5*d^4*e^6 - 20*a^3*c^4*d^2*
e^8 + a^4*c^3*e^10)/(a*c^10*d^20 + 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 + 120*a^4*c^7*d^14*e^6 + 210*a^5*
c^6*d^12*e^8 + 252*a^6*c^5*d^10*e^10 + 210*a^7*c^4*d^8*e^12 + 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 + 10*
a^10*c*d^2*e^18 + a^11*e^20)))/(a*c^5*d^10 + 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 + 10*a^4*c^2*d^4*e^6 + 5*a
^5*c*d^2*e^8 + a^6*e^10))) + 3*(2*c^2*d^5*x*e + c^2*d^6 + 4*a*c*d^3*x*e^3 + a^2*x^2*e^6 + 2*a^2*d*x*e^5 + (2*a
*c*d^2*x^2 + a^2*d^2)*e^4 + (c^2*d^4*x^2 + 2*a*c*d^4)*e^2)*sqrt(-(c^4*d^5 - 10*a*c^3*d^3*e^2 + 5*a^2*c^2*d*e^4
- (a*c^5*d^10 + 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 + 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 + a^6*e^10)*sqr
t(-(25*c^7*d^8*e^2 - 100*a*c^6*d^6*e^4 + 110*a^2*c^5*d^4*e^6 - 20*a^3*c^4*d^2*e^8 + a^4*c^3*e^10)/(a*c^10*d^20
+ 10*a^2*c^9*d^18*e^2 + 45*a^3*c^8*d^16*e^4 + 120*a^4*c^7*d^14*e^6 + 210*a^5*c^6*d^12*e^8 + 252*a^6*c^5*d^10*
e^10 + 210*a^7*c^4*d^8*e^12 + 120*a^8*c^3*d^6*e^14 + 45*a^9*c^2*d^4*e^16 + 10*a^10*c*d^2*e^18 + a^11*e^20)))/(
a*c^5*d^10 + 5*a^2*c^4*d^8*e^2 + 10*a^3*c^3*d^6*e^4 + 10*a^4*c^2*d^4*e^6 + 5*a^5*c*d^2*e^8 + a^6*e^10))*log((5
*c^4*d^4*e - 10*a*c^3*d^2*e^3 + a^2*c^2*e^5)*sqrt(x*e + d) + (15*a*c^4*d^6*e^2 - 35*a^2*c^3*d^4*e^4 + 13*a^3*c
^2*d^2*e^6 - a^4*c*e^8 - (a*c^6*d^13 + 2*a^2*c^5*d^11*e^2 - 5*a^3*c^4*d^9*e^4 - 20*a^4*c^3*d^7*e^6 - 25*a^5*c^
2*d^5*e^8 - 14*a^6*c*d^3*e^10 - 3*a^7*d*e^12)*sqrt(-(25*c^7*d^8*e^2 - 100*a*c^6*d^6*e^4 + 110*a^2*c^5*d^4*e^6
- 20*a^3*c^4*d^2*e^8 + a^4*c^3*e^10)/(a*c^10*d^...
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + c x^{2}\right ) \left (d + e x\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x+d)**(5/2)/(c*x**2+a),x)
[Out]
Integral(1/((a + c*x**2)*(d + e*x)**(5/2)), x)
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Giac [A]
time = 3.28, size = 506, normalized size = 0.69 \begin {gather*} -\frac {\sqrt {-a c} {\left | c \right |} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {c^{3} d^{5} + 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4} + \sqrt {{\left (c^{3} d^{5} + 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )}^{2} - {\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} {\left (c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )}}}{c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}}}}\right )}{{\left (a c d^{2} + 2 \, \sqrt {-a c} a d e - a^{2} e^{2}\right )} \sqrt {-c^{2} d - \sqrt {-a c} c e}} + \frac {\sqrt {-a c} {\left | c \right |} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-\frac {c^{3} d^{5} + 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4} - \sqrt {{\left (c^{3} d^{5} + 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )}^{2} - {\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} {\left (c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )}}}{c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}}}}\right )}{{\left (a c d^{2} - 2 \, \sqrt {-a c} a d e - a^{2} e^{2}\right )} \sqrt {-c^{2} d + \sqrt {-a c} c e}} - \frac {2 \, {\left (6 \, {\left (x e + d\right )} c d e + c d^{2} e + a e^{3}\right )}}{3 \, {\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} {\left (x e + d\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(e*x+d)^(5/2)/(c*x^2+a),x, algorithm="giac")
[Out]
-sqrt(-a*c)*abs(c)*arctan(sqrt(x*e + d)/sqrt(-(c^3*d^5 + 2*a*c^2*d^3*e^2 + a^2*c*d*e^4 + sqrt((c^3*d^5 + 2*a*c
^2*d^3*e^2 + a^2*c*d*e^4)^2 - (c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6)*(c^3*d^4 + 2*a*c^2*d^2*e
^2 + a^2*c*e^4)))/(c^3*d^4 + 2*a*c^2*d^2*e^2 + a^2*c*e^4)))/((a*c*d^2 + 2*sqrt(-a*c)*a*d*e - a^2*e^2)*sqrt(-c^
2*d - sqrt(-a*c)*c*e)) + sqrt(-a*c)*abs(c)*arctan(sqrt(x*e + d)/sqrt(-(c^3*d^5 + 2*a*c^2*d^3*e^2 + a^2*c*d*e^4
- sqrt((c^3*d^5 + 2*a*c^2*d^3*e^2 + a^2*c*d*e^4)^2 - (c^3*d^6 + 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 + a^3*e^6)*
(c^3*d^4 + 2*a*c^2*d^2*e^2 + a^2*c*e^4)))/(c^3*d^4 + 2*a*c^2*d^2*e^2 + a^2*c*e^4)))/((a*c*d^2 - 2*sqrt(-a*c)*a
*d*e - a^2*e^2)*sqrt(-c^2*d + sqrt(-a*c)*c*e)) - 2/3*(6*(x*e + d)*c*d*e + c*d^2*e + a*e^3)/((c^2*d^4 + 2*a*c*d
^2*e^2 + a^2*e^4)*(x*e + d)^(3/2))
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Mupad [B]
time = 2.57, size = 2500, normalized size = 3.40 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/((a + c*x^2)*(d + e*x)^(5/2)),x)
[Out]
- ((2*e)/(3*(a*e^2 + c*d^2)) + (4*c*d*e*(d + e*x))/(a*e^2 + c*d^2)^2)/(d + e*x)^(3/2) - atan((((d + e*x)^(1/2)
*(320*a^2*c^11*d^12*e^6 - 16*c^13*d^16*e^2 - 16*a^8*c^5*e^18 + 1024*a^3*c^10*d^10*e^8 + 1440*a^4*c^9*d^8*e^10
+ 1024*a^5*c^8*d^6*e^12 + 320*a^6*c^7*d^4*e^14) + (-(a^2*e^5*(-a^3*c^3)^(1/2) + a*c^4*d^5 + 5*a^3*c^2*d*e^4 -
10*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(-a^3*c^3)^(1/2) - 10*a*c*d^2*e^3*(-a^3*c^3)^(1/2))/(4*(a^7*e^10 + a^2*c^5*d^
10 + 5*a^6*c*d^2*e^8 + 5*a^3*c^4*d^8*e^2 + 10*a^4*c^3*d^6*e^4 + 10*a^5*c^2*d^4*e^6)))^(1/2)*(96*a*c^13*d^18*e^
3 - (d + e*x)^(1/2)*(-(a^2*e^5*(-a^3*c^3)^(1/2) + a*c^4*d^5 + 5*a^3*c^2*d*e^4 - 10*a^2*c^3*d^3*e^2 + 5*c^2*d^4
*e*(-a^3*c^3)^(1/2) - 10*a*c*d^2*e^3*(-a^3*c^3)^(1/2))/(4*(a^7*e^10 + a^2*c^5*d^10 + 5*a^6*c*d^2*e^8 + 5*a^3*c
^4*d^8*e^2 + 10*a^4*c^3*d^6*e^4 + 10*a^5*c^2*d^4*e^6)))^(1/2)*(64*a*c^14*d^21*e^2 + 64*a^11*c^4*d*e^22 + 640*a
^2*c^13*d^19*e^4 + 2880*a^3*c^12*d^17*e^6 + 7680*a^4*c^11*d^15*e^8 + 13440*a^5*c^10*d^13*e^10 + 16128*a^6*c^9*
d^11*e^12 + 13440*a^7*c^8*d^9*e^14 + 7680*a^8*c^7*d^7*e^16 + 2880*a^9*c^6*d^5*e^18 + 640*a^10*c^5*d^3*e^20) -
32*a^10*c^4*e^21 + 736*a^2*c^12*d^16*e^5 + 2432*a^3*c^11*d^14*e^7 + 4480*a^4*c^10*d^12*e^9 + 4928*a^5*c^9*d^10
*e^11 + 3136*a^6*c^8*d^8*e^13 + 896*a^7*c^7*d^6*e^15 - 128*a^8*c^6*d^4*e^17 - 160*a^9*c^5*d^2*e^19))*(-(a^2*e^
5*(-a^3*c^3)^(1/2) + a*c^4*d^5 + 5*a^3*c^2*d*e^4 - 10*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(-a^3*c^3)^(1/2) - 10*a*c*
d^2*e^3*(-a^3*c^3)^(1/2))/(4*(a^7*e^10 + a^2*c^5*d^10 + 5*a^6*c*d^2*e^8 + 5*a^3*c^4*d^8*e^2 + 10*a^4*c^3*d^6*e
^4 + 10*a^5*c^2*d^4*e^6)))^(1/2)*1i + ((d + e*x)^(1/2)*(320*a^2*c^11*d^12*e^6 - 16*c^13*d^16*e^2 - 16*a^8*c^5*
e^18 + 1024*a^3*c^10*d^10*e^8 + 1440*a^4*c^9*d^8*e^10 + 1024*a^5*c^8*d^6*e^12 + 320*a^6*c^7*d^4*e^14) - (-(a^2
*e^5*(-a^3*c^3)^(1/2) + a*c^4*d^5 + 5*a^3*c^2*d*e^4 - 10*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(-a^3*c^3)^(1/2) - 10*a
*c*d^2*e^3*(-a^3*c^3)^(1/2))/(4*(a^7*e^10 + a^2*c^5*d^10 + 5*a^6*c*d^2*e^8 + 5*a^3*c^4*d^8*e^2 + 10*a^4*c^3*d^
6*e^4 + 10*a^5*c^2*d^4*e^6)))^(1/2)*((d + e*x)^(1/2)*(-(a^2*e^5*(-a^3*c^3)^(1/2) + a*c^4*d^5 + 5*a^3*c^2*d*e^4
- 10*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(-a^3*c^3)^(1/2) - 10*a*c*d^2*e^3*(-a^3*c^3)^(1/2))/(4*(a^7*e^10 + a^2*c^5
*d^10 + 5*a^6*c*d^2*e^8 + 5*a^3*c^4*d^8*e^2 + 10*a^4*c^3*d^6*e^4 + 10*a^5*c^2*d^4*e^6)))^(1/2)*(64*a*c^14*d^21
*e^2 + 64*a^11*c^4*d*e^22 + 640*a^2*c^13*d^19*e^4 + 2880*a^3*c^12*d^17*e^6 + 7680*a^4*c^11*d^15*e^8 + 13440*a^
5*c^10*d^13*e^10 + 16128*a^6*c^9*d^11*e^12 + 13440*a^7*c^8*d^9*e^14 + 7680*a^8*c^7*d^7*e^16 + 2880*a^9*c^6*d^5
*e^18 + 640*a^10*c^5*d^3*e^20) - 32*a^10*c^4*e^21 + 96*a*c^13*d^18*e^3 + 736*a^2*c^12*d^16*e^5 + 2432*a^3*c^11
*d^14*e^7 + 4480*a^4*c^10*d^12*e^9 + 4928*a^5*c^9*d^10*e^11 + 3136*a^6*c^8*d^8*e^13 + 896*a^7*c^7*d^6*e^15 - 1
28*a^8*c^6*d^4*e^17 - 160*a^9*c^5*d^2*e^19))*(-(a^2*e^5*(-a^3*c^3)^(1/2) + a*c^4*d^5 + 5*a^3*c^2*d*e^4 - 10*a^
2*c^3*d^3*e^2 + 5*c^2*d^4*e*(-a^3*c^3)^(1/2) - 10*a*c*d^2*e^3*(-a^3*c^3)^(1/2))/(4*(a^7*e^10 + a^2*c^5*d^10 +
5*a^6*c*d^2*e^8 + 5*a^3*c^4*d^8*e^2 + 10*a^4*c^3*d^6*e^4 + 10*a^5*c^2*d^4*e^6)))^(1/2)*1i)/(((d + e*x)^(1/2)*(
320*a^2*c^11*d^12*e^6 - 16*c^13*d^16*e^2 - 16*a^8*c^5*e^18 + 1024*a^3*c^10*d^10*e^8 + 1440*a^4*c^9*d^8*e^10 +
1024*a^5*c^8*d^6*e^12 + 320*a^6*c^7*d^4*e^14) - (-(a^2*e^5*(-a^3*c^3)^(1/2) + a*c^4*d^5 + 5*a^3*c^2*d*e^4 - 10
*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(-a^3*c^3)^(1/2) - 10*a*c*d^2*e^3*(-a^3*c^3)^(1/2))/(4*(a^7*e^10 + a^2*c^5*d^10
+ 5*a^6*c*d^2*e^8 + 5*a^3*c^4*d^8*e^2 + 10*a^4*c^3*d^6*e^4 + 10*a^5*c^2*d^4*e^6)))^(1/2)*((d + e*x)^(1/2)*(-(
a^2*e^5*(-a^3*c^3)^(1/2) + a*c^4*d^5 + 5*a^3*c^2*d*e^4 - 10*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(-a^3*c^3)^(1/2) - 1
0*a*c*d^2*e^3*(-a^3*c^3)^(1/2))/(4*(a^7*e^10 + a^2*c^5*d^10 + 5*a^6*c*d^2*e^8 + 5*a^3*c^4*d^8*e^2 + 10*a^4*c^3
*d^6*e^4 + 10*a^5*c^2*d^4*e^6)))^(1/2)*(64*a*c^14*d^21*e^2 + 64*a^11*c^4*d*e^22 + 640*a^2*c^13*d^19*e^4 + 2880
*a^3*c^12*d^17*e^6 + 7680*a^4*c^11*d^15*e^8 + 13440*a^5*c^10*d^13*e^10 + 16128*a^6*c^9*d^11*e^12 + 13440*a^7*c
^8*d^9*e^14 + 7680*a^8*c^7*d^7*e^16 + 2880*a^9*c^6*d^5*e^18 + 640*a^10*c^5*d^3*e^20) - 32*a^10*c^4*e^21 + 96*a
*c^13*d^18*e^3 + 736*a^2*c^12*d^16*e^5 + 2432*a^3*c^11*d^14*e^7 + 4480*a^4*c^10*d^12*e^9 + 4928*a^5*c^9*d^10*e
^11 + 3136*a^6*c^8*d^8*e^13 + 896*a^7*c^7*d^6*e^15 - 128*a^8*c^6*d^4*e^17 - 160*a^9*c^5*d^2*e^19))*(-(a^2*e^5*
(-a^3*c^3)^(1/2) + a*c^4*d^5 + 5*a^3*c^2*d*e^4 - 10*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(-a^3*c^3)^(1/2) - 10*a*c*d^
2*e^3*(-a^3*c^3)^(1/2))/(4*(a^7*e^10 + a^2*c^5*d^10 + 5*a^6*c*d^2*e^8 + 5*a^3*c^4*d^8*e^2 + 10*a^4*c^3*d^6*e^4
+ 10*a^5*c^2*d^4*e^6)))^(1/2) - ((d + e*x)^(1/2)*(320*a^2*c^11*d^12*e^6 - 16*c^13*d^16*e^2 - 16*a^8*c^5*e^18
+ 1024*a^3*c^10*d^10*e^8 + 1440*a^4*c^9*d^8*e^10 + 1024*a^5*c^8*d^6*e^12 + 320*a^6*c^7*d^4*e^14) + (-(a^2*e^5*
(-a^3*c^3)^(1/2) + a*c^4*d^5 + 5*a^3*c^2*d*e^4 - 10*a^2*c^3*d^3*e^2 + 5*c^2*d^4*e*(-a^3*c^3)^(1/2) - 10*a*c*d^
2*e^3*(-a^3*c^3)^(1/2))/(4*(a^7*e^10 + a^2*c^5*d^10 + 5*a^6*c*d^2*e^8 + 5*a^3*c^4*d^8*e^2 + 10*a^4*c^3*d^6*e^4
+ 10*a^5*c^2*d^4*e^6)))^(1/2)*(96*a*c^13*d^18*...
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